Theorem lejoin1 18441

Description: A join's first argument is less than or equal to the join. (Contributed by NM, 16-Sep-2011.)

Hypotheses

Ref	Expression
joinval2.b	⊢ 𝐵 = (Base‘𝐾)
joinval2.l	⊢ ≤ = (le‘𝐾)
joinval2.j	⊢ ∨ = (join‘𝐾)
joinval2.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)
joinval2.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
joinval2.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
joinlem.e	⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∨ )

Assertion

Ref	Expression
lejoin1	⊢ (𝜑 → 𝑋 ≤ (𝑋 ∨ 𝑌))

Proof of Theorem lejoin1

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		joinval2.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		joinval2.l	. . 3 ⊢ ≤ = (le‘𝐾)
3		joinval2.j	. . 3 ⊢ ∨ = (join‘𝐾)
4		joinval2.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
5		joinval2.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		joinval2.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
7		joinlem.e	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∨ )
8	1, 2, 3, 4, 5, 6, 7	joinlem 18440	. 2 ⊢ (𝜑 → ((𝑋 ≤ (𝑋 ∨ 𝑌) ∧ 𝑌 ≤ (𝑋 ∨ 𝑌)) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → (𝑋 ∨ 𝑌) ≤ 𝑧)))
9	8	simplld 779	1 ⊢ (𝜑 → 𝑋 ≤ (𝑋 ∨ 𝑌))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1568   ∈ wcel 2150  ∀wral 3086  ⟨cop 4600   class class class wbr 5114  dom cdm 5665  ‘cfv 6540  (class class class)co 7414  Basecbs 17272  lecple 17320  joincjn 18370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7371  df-ov 7417  df-oprab 7418  df-lub 18403  df-join 18405

This theorem is referenced by:  joinle  18443  latlej1  18507